This set of Multiple Choice Questions & Answers (MCQs) focuses on “Robotics – Set 5”.
1. The transformation matrix is obtained by which off the following?
a) Product of rotation matrix and translation matrix
b) Sum of rotation matrix and translation matrix
c) Difference between the rotation matrix and translation matrix
d) Division of rotation matrix by translation matrix
View Answer
Explanation: The homogeneous transformation matrix uses the original coordinate frame to describe both rotation and translation. The transformation matrix is found by the product of rotation matrix and translation matrix.
2. How many sub-transformation matrices does a homogeneous transformation matrix have?
a) 2
b) 1
c) 4
d) 3
View Answer
Explanation: The homogeneous representation is a special case of homogeneous coordinates, which have been extensively used in the field of computer graphics. It consists of 4 sub-matrices namely the perspective matrix, scale factor, rotational matrix and transformation matrix.
H = \(\begin{bmatrix}
R_{3×3} & d_{3×1}\\
f_{1×3} & S_{1×1}
\end{bmatrix}\)
Where, R3×3 = Rotation
f1×3 = perspective
d3×1 = Translation
s1×1 = Scale factor
3. A transformation of the form given in Equation p0 = R0 1 * p1 + d0 1 is said to define a rigid motion if R is orthogonal. True or False?
a) True
b) False
View Answer
Explanation: Definition of a rigid motion includes reflections when detR = −1. If we have the two rigid motions, then their composition defines a third rigid motion. Simply by substituting the terms in the general equation and on comparison we can conclude that a transformation of the form given in Equation p0 = R0 1 * p1 + d0 1 is said to define a rigid motion if R is orthogonal.
4. Which of the following denotes composite transformation of matrices between frame {3} and frame {1} ?
a) 1T3 = 1T2 * 2T3
b) 1T3 = 1T2 + 2T3
c) 1T3 = 1T2 – 2T3
d) 1T3 = 1T2 / 2T3
View Answer
Explanation: The individual homogeneous matrices should be multiplied together to obtain composite homogeneous transformation matrix. The order of multiplication is fixed as it is not commutative.
5. The vector ṽ with coordinates v0 = (0,1,1)T is rotated about y0 by 90 degrees. Then what are the resulting coordinates of vector ṽ1?
Rotating a vector about y0
a) (0,1,1)T
b) (1,1,0)T
c) (1,0,1)T
d) (1,1,1)T
View Answer
Explanation: We know that, ṽ01 = Ry,90 * v0
\(\begin{pmatrix}
0&0&1\\
0&1&0\\
1&0&0
\end{pmatrix}*(0,1,1)^T\) = (1,1,0)T
6. Which of the following represents a homogeneous transformation matrix H that represents a rotation of α degrees about the current x-axis followed by a translation of b units along the current x-axis, followed by a translation of d units along the current z-axis, followed by a rotation of θ degrees about the current z-axis?
a) \(\begin{bmatrix}
1 & -sθ & 0 & b\\
cα sα & cα cθ & -sα & -sα d\\
sα sθ & sα cθ & cα & cα d\\
0 & 0 & 1 & 1\\
\end{bmatrix}\)
b) \(\begin{bmatrix}
cθ & -sθ & 0 & b\\
cα sα & cα cθ & -sα & -sα d\\
sα sθ & sα cθ & cα & cα d\\
0 & 0 & 0 & 1\\
\end{bmatrix}\)
c) \(\begin{bmatrix}
cθ & -sθ & 0 & b\\
cα sα & 0 & -sα & 0\\
sα sθ & sα cθ & cα & cα d\\
0 & 0 & 0 & 1\\
\end{bmatrix}\)
d) \(\begin{bmatrix}
cθ & -sθ & 0 & b\\
0 & cα cθ & -sα & -sα d\\
sα sθ & sα cθ & cα & 1\\
0 & 0 & 0 & 1\\
\end{bmatrix}\)
View Answer
Explanation: H = Rotx, α * Transx, b * Transz,d * Rotz, θ
=\(\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & cα & -sα & 0\\
0 & sα & cα & 0\\
0 & 0 & 0 & 1\\
\end{bmatrix} = \begin{bmatrix}
1 & 0 & 0 & b\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
\end{bmatrix} = \begin{bmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & d\\
0 & 0 & 0 & 1\\
\end{bmatrix} = \begin{bmatrix}
cθ & -sθ & 0 & 0\\
sθ & cθ & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
\end{bmatrix}\)
= \(\begin{bmatrix}
cθ & -sθ & 0 & b\\
cα sα & cα cθ & -sα & -sα d\\
sα sθ & sα cθ & cα & cα d\\
0 & 0 & 0 & 1\\
\end{bmatrix}\)
7. What is the range of global scaling parameter for reducing purpose?
a) Global scaling parameter > 1
b) Global scaling parameter < 1
c) Global scaling parameter = 1
d) 0 < Global scaling parameter < 1
View Answer
Explanation: The scale factor has non zero positive value and is called as global scaling parameter. Global scaling parameter > 1 is useful for reducing and 0 < Global scaling parameter < 1 is useful for enlarging. For robotic study Global scaling parameter = 1 is used.
8. While describing a transformation matrix, the sub-matrix i.e, perspective transformation matrix is determined as nxm matrix. Which of the following indicates the correct value of n, m?
a) 1,3
b) 3,3
c) 3,1
d) 1,1
View Answer
Explanation: Perspective transformation matrix is a 1×3 matrix. It is useful in vision systems and is set to zero vector whenever no perspective views are involved.
9. While describing a transformation matrix, one of its sub-matrix i.e, translation matrix is determined as nxm matrix. Which of the following indicates the correct value of n, m?
a) 1,3
b) 3,3
c) 3,1
d) 1,1
View Answer
Explanation: Translation matrix is 3×1 matrix. Translation matrix is used to denote the displacement of a point in a frame along any one corresponding axes x, y, or z axis.
H = \(\begin{bmatrix}
R_{3×3} & d_{3×1}\\
f_{1×3} & S_{1×1}
\end{bmatrix}\)
Where, R3×3 = Rotation
f1×3 = perspective
d3×1 = Translation
s1×1 = Scale factor
10. If 1T2 is denoted as
\(\begin{bmatrix}
1^R_2 & 1^D_2\\
000 & 1
\end{bmatrix}\)
Then how will the inverse of 1T2 be denoted as ?
a) 2T1 = 1T2 = \(\begin{bmatrix}
1^R_2 & 1^D_2\\
000 & 1
\end{bmatrix}\)
b) 2T1 =[1T2]‘ = \(\begin{bmatrix}
(1^R_2)^T & -(1^R_2)^T 1^D_2\\
000 & 1
\end{bmatrix}\)
c) identity matrix
d) inverse cannot be determined from 1T2
View Answer
Explanation: We can write 1T2 as \(\begin{bmatrix}
1^R_2 & 1^D_2\\
000 & 1
\end{bmatrix}\)
Similarly, 2T1 as \(\begin{bmatrix}
1^R_2 & 1^D_2\\
000 & 1
\end{bmatrix}\)
We know that for Rotation matrix, 2R1 = [1R2]T.
By mapping the points in frame {2} to frame {1} and comparing the rotational matrices we get
2T1=[1T2]‘ = \(\begin{bmatrix}
(1^R_2)^T & -(1^R_2)^T 1^D_2\\
000 & 1
\end{bmatrix}\)
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